Multi-agent coordination dilemmas expose a fundamental tension between individual optimization and collective welfare, yet characterizing such coordination requires metrics sensitive to temporal structure and collective dynamics. As a diagnostic testbed, we study a BoE-derived multi-agent variant of the Battle of the Exes, formalizing it as a Markov game in which turn-taking emerges as a periodic coordination regime. Conventional outcome-based metrics (e.g., efficiency and min/max fairness) are temporally blind (they cannot distinguish structured alternation from monopolistic or random access patterns) and fairness ratios lose discriminative power as n grows, obscuring inequities. To address this limitation, we introduce Perfect Alternation (PA) as a reference coordination regime and propose six novel Alternation (ALT) metrics designed as temporally sensitive observables of coordination quality. Using Q-learning agents as a minimal adaptive diagnostic baseline, and comparing against random-policy null processes, we uncover a clear measurement failure: despite exhibiting deceptively high traditional metrics (e.g., reward fairness often exceeding 0.9), learned policies perform up to 81% below random baselines under ALT-variant evaluation, a deficit already present in the two-agent case and intensifying as n grows. These results demonstrate, in this setting, that high aggregate payoffs can coexist with poor temporal coordination, and that conventional metrics may severely mischaracterize emergent dynamics. Our findings underscore the necessity of temporally aware observables for analyzing coordination in multi-agent games and highlight random-policy baselines as essential null processes for interpreting coordination outcomes relative to chance-level behavior.

翻译：多智能体协调困境暴露了个体优化与集体福祉之间的根本张力，然而表征此类协调需要能感知时序结构和集体动态的度量方法。作为诊断测试平台，我们研究了从"伴侣争夺博弈"衍生的多智能体变体，将其形式化为马尔可夫博弈，其中轮流行为作为周期性协调机制涌现。传统基于结果的度量（如效率与最大/最小公平性）存在时序盲区（无法区分结构化交替模式与垄断性或随机访问模式），且公平比率随智能体数量增加而丧失区分能力，从而模糊了不平等性。为解决这一局限，我们引入"完美交替"作为参考协调机制，并提出六种新型交替度量作为协调质量的时序敏感可观测指标。采用Q学习智能体作为最小自适应诊断基准，并与随机策略零假设过程对比，我们发现了显著的测量失效：尽管传统指标呈现欺骗性高值（例如奖励公平性常超过0.9），但学习策略在交替变体评估下表现可比随机基准低81%，这种缺陷在双智能体案例中已然存在，并随智能体数量增加而加剧。这些结果表明，在该设定下，高聚合收益可与不良时序协调共存，传统度量可能严重误判涌现动力学。我们的发现凸显了分析多智能体博弈协调时需采用时序感知可观测指标，并强调随机策略基准作为解读协调结果相对于随机水平行为的必需零假设过程。